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Complete quenching phenomenon for a parabolic p-Laplacian equation with a weighted absorption
- Liping Zhu^{1}Email author
https://doi.org/10.1186/s13660-018-1841-5
© The Author(s) 2018
- Received: 8 February 2018
- Accepted: 6 September 2018
- Published: 20 September 2018
Abstract
Throughout this paper, we mainly consider the parabolic p-Laplacian equation with a weighted absorption \(u_{t}-\operatorname{div} (|\nabla u|^{p-2}\nabla u )=-\lambda |x|^{\alpha} {\chi}_{\{u>0\}}u^{-\beta}\) in a bounded domain \(\Omega\subseteq\mathbb{R}^{n}\) (\(n\geq1\)) with Lipschitz continuous boundary subject to homogeneous Dirichlet boundary condition. Here \(\lambda>0\) and \(\alpha>-n\) are parameters, and \(\beta\in(0,1)\) is a given constant. Under the assumptions \(u_{0}\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)\), \(u_{0}\geq0\) a.e. in Ω, we can establish conditions of local and global in time existence of nonnegative solutions, and show that every global solution completely quenches in finite time a.e. in Ω. Moreover, we give some numerical experiments to illustrate the theoretical results.
Keywords
- Parabolic p-Laplacian
- Complete quenching
- Weighted absorption
MSC
- 35D30
- 35K55
- 35K92
1 Introduction
Moreover, Guo [26] studied the problem (1.4) for \(f(x)=|x|^{\alpha}\), \(\alpha>0\), and Ω being the unit ball in \(\mathbb{R}^{n}\) (\(n\geq2\)). Under certain conditions of \(\lambda, n\) and α, Guo showed the stability of the minimal compact stationary solution and the instability of the singular stationary solution of (1.4), respectively. Guo and Wei [29] studied the Cauchy problem with a singular nonlinearity \(u_{t}=\Delta u-u^{-\nu }\) with \(\nu>0\) and proved that the problem has a global classical solution, and studied the properties of positive radial solutions of the steady state. More generally, Castorina et al. [5] studied the p-MEMS equation \(-\Delta_{p}u={\lambda}/{(1-u)^{2}}\) in a ball and proved the uniqueness of semi-stable solutions and stability of minimal solutions for \(1< p\leq2\).
There are some recent works on local and global existence, gradient estimates, blowup and extinction of the p-Laplacian equations. We refer to [32, 35, 44, 45] for the nonlinear absorption and source, nonlinear gradient absorption or source, and [9, 10, 22] for singular absorptions. Also, we refer to [46, 47] for the semilinear equations with an exponential source. When \(\alpha=0\), equation (1.1) is known as a limit model of a class of problems arising in Chemical Engineering corresponding to enzymatic kinetics and hetergeneous catalyst of Langmuir–Hinshelwood type, see [3, 9, 12, 15, 22, 39, 43] and references therein. Under the Dirichlet boundary condition, problem (1.1) of \(p=2\) has been studied by many authors, we refer to [14, 19, 30] and the references therein. The Cauchy problem for equation (1.1) was studied by Phillips [39]. Winkler [42] studied the nondivergent parabolic equations with singular absorption. Under certain conditions, Giacomoni et al. [22] showed that problem (1.1) has a global in time bounded weak solution. Moreover, every weak solution u completely quenches in a finite time \(T_{\ast}\), i.e., \(u(\cdot,t)=0\) a.e. in Ω for all t beyond \(T_{\ast}\).
Due to the singular absorption, the solution u of (1.1) may quench in finite time on one set with nonzero measure, even if the initial datum is strictly positive (see [11–13, 37]). Davila and Montenegro [11–13] have studied the semilinear problem (1.1) with \(p=2\) and \(\alpha=0\) under the assumptions \(u_{0}\geq0\) a.e. in Ω and \(u_{0}\in L^{\infty}(\Omega)\cap C(\Omega)\). Moreover, under certain stronger conditions on \(u_{0}\), Montenegro [37] showed that the solution u of (1.1) with \(p=2\) and \(\alpha =0\) may quench completely.
2 Definition of weak solutions and main results
Definition 2.1
- 1.
\(u\in \mathscr {U}\cap C ([0,T];L^{2}(\Omega) )\), \(u\geq0\) a.e. in \(Q_{T}\);
- 2.\(|x|^{\alpha}\chi_{\{u>0\}}u^{-\beta}\varphi\in L^{1}(Q_{T})\) holds for every test function \(\varphi\in \mathscr {U}\), and$$\int_{Q_{T}}\partial_{t}u\cdot\varphi \,dz+ \int_{Q_{T}}|\nabla u|^{p-2}\nabla u\cdot\nabla\varphi \,dz +\lambda \int_{Q_{T}}|x|^{\alpha}\chi_{\{u>0\}}u^{-\beta} \varphi \,dz=0; $$
- 3.
\(u(x,0)=u_{0}\) a.e. in Ω.
Next, we give the main results of this paper.
Theorem 2.1
Theorem 2.2
3 Global weak solutions
For problem (1.1) with \(\alpha=0\), the existence of local in time weak solutions can be obtained by studying the regularization equation and proving the uniform gradient estimates, and then passing the parameter to a limit. We refer to [9, 10, 22] for the details of proof, and Theorem 2.1 can be derived in a similar manner to [22, Theorem 2.1] (see also [10, Theorem 2] for the degenerate case of \(p>2\) and \(n=1\)).
Here we are mainly interested in the asymptotic behavior of nonnegative and global solutions of the weighted problem (1.1). However, the equation is singular at \(x=0\) for \(-n<\alpha<0\). In fact, the solutions can be approximated, if necessary, by those satisfying the regularized equation \(u_{t}-\Delta_{p} u=-\lambda(|x|+\epsilon)^{\alpha }{\chi}_{\{u>0\}}u^{-\beta}\) with the same initial-boundary value conditions and taking the limit \(\epsilon\rightarrow0^{+}\).
Theorem 3.1
Assume \(u_{0}\in L^{2}(\Omega)\), then (1.1) has a global in time weak solution if \(\alpha>\max \{-\frac{n(p+\beta-1)}{p},-\frac {n}{2} \}\), \(\lambda<\frac{\lambda_{1}p}{1-\beta}\).
Proof
We use the classical Faedo–Galerkin method for the parabolic equations (see [2, 34]) to prove this theorem. Here we just give a brief proof.
Theorem 3.2
Assume that \(u_{0}\in L^{\infty}\), \(u_{0}\geq0\) a.e. in Ω, then there exist \(M>0\) and \(T^{\ast}>0\) such that a solution v of (1.1) satisfies \(0\leq v\leq M\) a.e. in \(Q_{T}\) for \(T< T^{\ast}\).
Proof
Theorem 3.3
Let the conditions of Theorem 3.2 be satisfied. Then the solution v of (1.1) is global in time. Moreover, for every \(T>0\), v satisfies \(0\leq v\leq M\) a.e. in \(Q_{T}\), where \(M=M (p,\|u_{0}\|_{\infty,\Omega},\lambda_{1} )>0\).
By Theorem 3.2, we easily conclude that Theorem 3.3 can be established. Also, by the regularization arguments as when proving Theorem 3.4 in [22], we can derive the following theorem of higher regularity of solutions to problem (1.1). Here we state these results and omit the details (cf. [22]).
Theorem 3.4
4 Complete quenching in finite time
In this section, following the idea of [16, 22] (see also the book [1]), we discuss the complete quenching phenomenon by using the energy methods and give the proof of Theorem 2.2. We here note that Díaz [16] has extended the energy method to the study of the free boundary generated by the solutions of more general semilinear or quasilinear parabolic problems of quenching type, which involve a negative power of the unknown in an equation like (1.1).
Define the energy function \(J(t)= \|u(t) \|^{2}_{2,\Omega}\). In the following, we first derive the energy equality and ordinary differential inequality satisfied by \(J(t)\).
To prove the differential inequality satisfied by \(J(t)\) in Lemma 4.2, we will make use of the interpolation inequality with weights of Gagliardo–Nirenberg type (see [33]) as follows.
Lemma 4.1
Lemma 4.2
Proof
Proof of Theorem 2.2
Now we will complete the proof of Theorem 2.2, which can be proved by the following lemma. □
Lemma 4.3
Proof
5 Numerical experiments
In this section, we give some numerical experiments which illustrate our theoretical results.
We consider the case of one space variable and mimic the numerical scheme in [41], and by the pdepe solver we convert equation (1.1) to ODEs using a second-order accurate spatial discretization based on a fixed interval of specified nodes. We refer the interested readers to [41], where the discretization method is described in detail.
We take \(\Omega=[0,5]\) and \(0=x_{1}< x_{2}<\cdots<x_{N}=5\) with \(N=10\). By calling the pdepe function in Matlab, we can obtain the figures of numerical solution for \(p=2\) and \(p=4\), respectively. We know the solution will be quenching completely in finite time, through Theorem 2.2.
According to Figs. 1–16, we find that, as \(u_{0}\) gets larger, the complete quenching time will be also longer. Moreover, from the figures of \(\alpha=-0.1, 0.6, 0.66\), we know that the complete quenching phenomenon will occur when α increases to some critical value, for example, \(\alpha\sim0.66\) in above numerical experiments.
Remark 5.1
In this section, we only show the complete quenching phenomenon of numerical solutions by choosing some special parameters of λ, β, α, p and certain initial data. In other words, the global weak solutions obtained in Theorem 2.2 are not unique, in general. When \(p=2\), \(\lambda=1\) and \(\alpha=0\) are taken in equation (1.1), Winkler [43] has shown that, for any n and β, the nonuniqueness holds at least for some nonnegative boundary and initial data. We suspect that similar results would still hold for the quasilinear equation (1.1). We leave it to the interested readers as an open question.
Declarations
Funding
Zhu was supported by the National Natural Science Foundation of China (Nos. 11401458, 11702206) and the grant of China Scholarship Council (No. 201607835015).
Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares that she has no competing interests.
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Authors’ Affiliations
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